Algebraic Formula Sheet

Algebraic Formula Sheet

Arithmetic Operations

ac + bc = c(a + b)

b ab a=

cc

a

b

a

=

c bc

a ac

=

b

b

c

Properties of Inequalities

If a < b then a + c < b + c and a - c < b - c ab

If a < b and c > 0 then ac < bc and < cc ab

If a < b and c < 0 then ac > bc and > cc

a c ad + bc +=

b d bd

a c ad - bc -=

c d bd

a-b b-a =

c-d d-c

a+b a b =+

c cc

a

ab + ac = b + c, a = 0

a

b ad

=

c

bc

d

Properties of Exponents

xnxm = xn+m

x0 = 1, x = 0

Properties of Absolute Value x if x 0

|x| = -x if x < 0

|x| 0

| - x| = |x|

|xy| = |x||y|

x |x| =

y |y|

|x + y| |x| + |y| Triangle Inequality |x - y| |x| - |y| Reverse Triangle Inequality

(xn)m = xnm

(xy)n = xnyn

n

xm =

1n

xm =

xn

1 m

-n

n

x

y

yn

=

=

y

x

xn

n

x

xn

y = yn

1 = xn x-n xn = xn-m xm

x-n

=

1 xn

Distance Formula Given two points, PA = (x1, y1) and PB = (x2, y2), the distance between the two can be found by:

d(PA, PB) = (x2 - x1)2 + (y2 - y1)2

Number Classifications Natural Numbers : N={1, 2, 3, 4, 5, . . .}

Whole Numbers : {0, 1, 2, 3, 4, 5, . . .}

Properties of Radicals

nx

=

1

xn

x nx n = y ny

n xy = n x n y

n xn = x, if n is odd

m

nx=

x mn

n xn = |x|, if n is even

Integers : Z={... ,-3, -2, -1, 0, 1, 2, 3, .. .}

Rationals : Q= All numbers that can be written as a fraction with an integer numerator and a

a nonzero integer denominator,

b Irrationals : {All numbers that cannot be expressed as the ratio of two integers, for example

5, 27, and }

Real Numbers : R={All numbers that are either a rational or an irrational number}

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Logarithms and Log Properties

Definition y = logb x is equivalent to x = by Example log2 16 = 4 because 24 = 16 Special Logarithms ln x = loge x natural log where e=2.718281828... log x = log10 x common log

xa + xb = x(a + b) x2 - y2 = (x + y)(x - y) x2 + 2xy + y2 = (x + y)2 x2 - 2xy + y2 = (x - y)2 x3 + 3x2y + 3xy2 + y3 = (x + y)3 x3 - 3x2y + 3xy2 - y3 = (x - y)3

Logarithm Properties

logb b = 1 logb bx = x ln ex = x

logb 1 = 0 blogb x = x eln x = x

logb (xk) = k logb x logb (xy) = logb x + logb y

x logb y = logb x - logb y

Factoring x3 + y3 = (x + y) x2 - xy + y2 x3 - y3 = (x - y) x2 + xy + y2 x2n - y2n = (xn - yn) (xn + yn) If n is odd then, xn - yn = (x - y) xn-1 + xn-2y + ... + yn-1 xn + yn = (x + y) xn-1 - xn-2y + xn-3y2... - yn-1

Linear Functions and Formulas

Examples of Linear Functions

y

y

y=x x

y=1 x

linear f unction 2

constant f unction

Constant Function

This graph is a horizontal line passing through the points (x, c) with slope m = 0 :

y = c or f (x) = c

Linear Function/Slope-intercept form

This graph is a line with slope m and y - intercept(0, b) :

y = mx + b or f (x) = mx + b

Slope (a.k.a Rate of Change)

The slope m of the line passing through the points (x1, y1) and (x2, y2) is : m = y = y2 - y1 = rise

x x2 - x1 run

Point-Slope form

The equation of the line passing through the point (x1, y1) with slope m is :

y = m(x - x1) + y1

Quadratic Functions and Formulas

Examples of Quadratic Functions

y

y

y = x2 x

y = -x2 x

parabola opening up

parabola opening down

Forms of Quadratic Functions

Standard Form

Vertex Form

y = ax2 + bx + c

or f (x) = ax2 + bx + c

y = a(x - h)2 + k

or f (x) = a(x - h)2 + k

This graph is a parabola that opens up if a > 0 or down if

a < 0 and has a vertex at

b

b

- ,f -

.

2a

2a

This graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertex at (h, k).

3

Quadratics and Solving for x

Quadratic Formula To solve ax2 + bx + c = 0, a = 0,

use :

-b ? b2 - 4ac

x=

.

2a

The Discriminant

The discriminant is the part of the quadratic equation under the radical, b2 - 4ac. We use

the discriminant to determine the number of real solutions of ax2 + bx + c = 0 as such :

Square Root Property

Let k be a nonnegative number. Then the solutions to the equation

x2 = k are given by x = ? k.

1. If b2 - 4ac > 0, there are two real solutions. 2. If b2 - 4ac = 0, there is one real solution. 3. If b2 - 4ac < 0, there are no real solutions.

Other Useful Formulas

Compound Interest

r nt A=P 1+

n where: P= principal of P dollars r= Interest rate (expressed in decimal form) n= number of times compounded per year t= time

Continuously Compounded Interest

A = P ert

where: P= principal of P dollars r= Interest rate (expressed in decimal form) t= time

Circle

(x - h)2 + (y - k)2 = r2

This graph is a circle with radius r and center (h, k).

Ellipse

(x - h)2 (y - k)2 a2 + b2 = 1

This graph is an ellipse with center (h, k) with vertices a units right/left from the center and vertices b units up/down from the center.

Hyperbola

(x - h)2 (y - k)2

-

=1

a2

b2

This graph is a hyperbola that opens

left and right, has center (h, k), vertices

(h ? a, k); foci (h ? c, k), where c comes from c2 = a2 + b2 and

asymptotes that pass through the center b

y = ? (x - h) + k. a

(y - k)2 (x - h)2

-

=1

a2

b2

This graph is a hyperbola that opens up and down, has center (h, k), vertices (h, k ? a); foci (h, k ? c), where c comes from c2 = a2 + b2 and asymptotes that pass through the center

a y = ? (x - h) + k.

b

Pythagorean Theorem

A triangle with legs a and b and hypotenuse c is a right triangle if and only if

a2 + b2 = c2

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